Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,22)(17,23)(18,33)(19,24)(20,28)(21,29)(26,41)(27,36)(30,44)(31,38)(32,40)(34,39)(35,43)(37,42);(1,2,3,4)(13,15,17,27,19,29,18,28)(14,20,33,21,24,36,23,25)(16,30,34,26,32,43,31,42)(22,37,38,35,40,41,39,44);(6,9)(10,12)(13,15,19,29)(14,21,24,25)(16,26,32,42)(17,28,18,27)(20,23,36,33)(22,37,40,41)(30,31,43,34)(35,38,44,39);(5,7,8,11)(6,10,9,12)(13,16,19,32)(14,22,24,40)(15,26,29,42)(17,31,18,34)(20,35,36,44)(21,37,25,41)(23,38,33,39)(27,30,28,43);(13,17,19,18)(14,23,24,33)(15,27,29,28)(16,31,32,34)(20,25,36,21)(22,38,40,39)(26,30,42,43)(35,41,44,37)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356452);